10 Bessel Functions Spherical Bessel Functions 10.48 Graphs 10.50 Wronskians and Cross-Products

§10.49 Explicit Formulas

Permalink:: http://dlmf.nist.gov/10.49

§10.49(i) Unmodified Functions
§10.49(ii) Modified Functions
§10.49(iii) Rayleigh’s Formulas
§10.49(iv) Sums or Differences of Squares

§10.49(i) Unmodified Functions

Notes:: When $\nu=n+\tfrac{1}{2}$ the asymptotic expansions (10.17.3)–(10.17.6) terminate, and as a consequence of the error bounds of §10.17(iv) they represent the left-hand sides exactly.
Keywords:: spherical Bessel functions
Referenced by:: ¶ ‣ §10.16, §10.49(ii), §10.61(v), §10.74(i), §11.4(i)
Permalink:: http://dlmf.nist.gov/10.49.i

Define $a_{k}(\nu)$ as in (10.17.1). Then

10.49.1 $a_{k}(n+\tfrac{1}{2})=\begin{cases}\dfrac{(n+k)!}{2^{k}k!(n-k)!},&k=0,1,\ldots% ,n,\\ 0,&k=n+1,n+2,\ldots.\end{cases}$

Symbols:: : : factorial, : integer, : nonnegative integer and $a_{k}(\nu)$ : expansion
Referenced by:: §10.49(ii), §10.50
Permalink:: http://dlmf.nist.gov/10.49.E1
Encodings:: TeX, pMML, png

10.49.2 $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)=\mathop{\sin\/}\nolimits% \!\left(z-\tfrac{1}{2}n\pi\right)\sum_{{k=0}}^{{\left\lfloor n/2\right\rfloor}% }(-1)^{k}\frac{a_{{2k}}(n+\tfrac{1}{2})}{z^{{2k+1}}}+\mathop{\cos\/}\nolimits% \!\left(z-\tfrac{1}{2}n\pi\right)\sum_{{k=0}}^{{\left\lfloor(n-1)/2\right% \rfloor}}(-1)^{k}\frac{a_{{2k+1}}(n+\tfrac{1}{2})}{z^{{2k+2}}}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\left\lfloor x\right\rfloor$ : floor of , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : expansion
A&S Ref:: 10.1.8
Referenced by:: §10.50, §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E2
Encodings:: TeX, pMML, png

10.49.3

$\mathop{\mathsf{j}_{{0}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\sin\/}% \nolimits z}{z},$

$\mathop{\mathsf{j}_{{1}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\sin\/}% \nolimits z}{z^{2}}-\frac{\mathop{\cos\/}\nolimits z}{z},$

$\mathop{\mathsf{j}_{{2}}\/}\nolimits\!\left(z\right)=\left(-\frac{1}{z}+\frac{% 3}{z^{3}}\right)\mathop{\sin\/}\nolimits z-\frac{3}{z^{2}}\mathop{\cos\/}% \nolimits z.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind and : complex variable
A&S Ref:: 10.1.11 (corrected)
Referenced by:: §10.49(iii), §10.56
Permalink:: http://dlmf.nist.gov/10.49.E3
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

10.49.4 $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)=-\mathop{\cos\/}\nolimits% \!\left(z-\tfrac{1}{2}n\pi\right)\sum_{{k=0}}^{{\left\lfloor n/2\right\rfloor}% }(-1)^{k}\frac{a_{{2k}}(n+\tfrac{1}{2})}{z^{{2k+1}}}+\mathop{\sin\/}\nolimits% \!\left(z-\tfrac{1}{2}n\pi\right)\sum_{{k=0}}^{{\left\lfloor(n-1)/2\right% \rfloor}}(-1)^{k}\frac{a_{{2k+1}}(n+\tfrac{1}{2})}{z^{{2k+2}}}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\left\lfloor x\right\rfloor$ : floor of , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : expansion
A&S Ref:: 10.1.9
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E4
Encodings:: TeX, pMML, png

10.49.5

$\mathop{\mathsf{y}_{{0}}\/}\nolimits\!\left(z\right)=-\frac{\mathop{\cos\/}% \nolimits z}{z},$

$\mathop{\mathsf{y}_{{1}}\/}\nolimits\!\left(z\right)=-\frac{\mathop{\cos\/}% \nolimits z}{z^{2}}-\frac{\mathop{\sin\/}\nolimits z}{z},$

$\mathop{\mathsf{y}_{{2}}\/}\nolimits\!\left(z\right)=\left(\frac{1}{z}-\frac{3% }{z^{3}}\right)\mathop{\cos\/}\nolimits z-\frac{3}{z^{2}}\mathop{\sin\/}% \nolimits z.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind and : complex variable
A&S Ref:: 10.1.12 (corrected)
Referenced by:: §10.50, §10.56
Permalink:: http://dlmf.nist.gov/10.49.E5
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

10.49.6 $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=e^{{iz}}\sum_{{% k=0}}^{n}i^{{k-n-1}}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}},$

Symbols:: : base of exponential function, $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : expansion
A&S Ref:: 10.1.16
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E6
Encodings:: TeX, pMML, png

10.49.7 $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=e^{{-iz}}\sum_{% {k=0}}^{n}(-i)^{{k-n-1}}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}.$

Symbols:: : base of exponential function, $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : expansion
A&S Ref:: 10.1.17 (corrected)
Permalink:: http://dlmf.nist.gov/10.49.E7
Encodings:: TeX, pMML, png

§10.49(ii) Modified Functions

Notes:: Use the same method as for §10.49(i), or combine the results of §10.49(i) with (10.47.12) and (10.47.13).
Keywords:: Bessel polynomials, spherical Bessel functions
Referenced by:: ¶ ‣ §10.39, §10.74(i), §11.4(i), §18.34(i), §18.34(iii), §3.5(viii)
Permalink:: http://dlmf.nist.gov/10.49.ii

Again, with $a_{k}(n+\tfrac{1}{2})$ as in (10.49.1),

10.49.8 $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}e^{% z}\sum_{{k=0}}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}+(-1)^{{n+1}}% \*\tfrac{1}{2}e^{{-z}}\sum_{{k=0}}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}.$

Symbols:: : base of exponential function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : expansion
A&S Ref:: 10.2.9 (modified)
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E8
Encodings:: TeX, pMML, png

10.49.9

$\mathop{{\mathsf{i}^{{(1)}}_{{0}}}\/}\nolimits\!\left(z\right)=\frac{\mathop{% \sinh\/}\nolimits z}{z},$

$\mathop{{\mathsf{i}^{{(1)}}_{{1}}}\/}\nolimits\!\left(z\right)=-\frac{\mathop{% \sinh\/}\nolimits z}{z^{2}}+\frac{\mathop{\cosh\/}\nolimits z}{z},$

$\mathop{{\mathsf{i}^{{(1)}}_{{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{1}{% z}+\frac{3}{z^{3}}\right)\mathop{\sinh\/}\nolimits z-\frac{3}{z^{2}}\mathop{% \cosh\/}\nolimits z.$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function and : complex variable
A&S Ref:: 10.2.13
Permalink:: http://dlmf.nist.gov/10.49.E9
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

10.49.10 $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}e^{% z}\sum_{{k=0}}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}+(-1)^{n}% \tfrac{1}{2}e^{{-z}}\sum_{{k=0}}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}.$

Symbols:: : base of exponential function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : expansion
A&S Ref:: 10.2.10 (modified)
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E10
Encodings:: TeX, pMML, png

10.49.11

$\mathop{{\mathsf{i}^{{(2)}}_{{0}}}\/}\nolimits\!\left(z\right)=\frac{\mathop{% \cosh\/}\nolimits z}{z},$

$\mathop{{\mathsf{i}^{{(2)}}_{{1}}}\/}\nolimits\!\left(z\right)=-\frac{\mathop{% \cosh\/}\nolimits z}{z^{2}}+\frac{\mathop{\sinh\/}\nolimits z}{z},$

$\mathop{{\mathsf{i}^{{(2)}}_{{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{1}{% z}+\frac{3}{z^{3}}\right)\mathop{\cosh\/}\nolimits z-\frac{3}{z^{2}}\mathop{% \sinh\/}\nolimits z.$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function and : complex variable
A&S Ref:: 10.2.14
Permalink:: http://dlmf.nist.gov/10.49.E11
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

10.49.12 $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi e^{{-z}}% \sum_{{k=0}}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{{k+1}}}.$

Symbols:: : base of exponential function, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : expansion
A&S Ref:: 10.2.15
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E12
Encodings:: TeX, pMML, png

10.49.13

$\mathop{\mathsf{k}_{{0}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi\frac{e^{{% -z}}}{z},$

$\mathop{\mathsf{k}_{{1}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi e^{{-z}}% \left(\frac{1}{z}+\frac{1}{z^{2}}\right),$

$\mathop{\mathsf{k}_{{2}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi e^{{-z}}% \left(\frac{1}{z}+\frac{3}{z^{2}}+\frac{3}{z^{3}}\right).$

Symbols:: : base of exponential function, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function and : complex variable
A&S Ref:: 10.2.17
Permalink:: http://dlmf.nist.gov/10.49.E13
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

$\sum_{{k=0}}^{n}a_{k}(n+\tfrac{1}{2})z^{{n-k}}$ is sometimes called the Bessel polynomial of degree . For a survey of properties of these polynomials and their generalizations see Grosswald (1978). See also §18.34, de Bruin et al. (1981a, b), and Dunster (2001c).

§10.49(iii) Rayleigh’s Formulas

Notes:: For the first of(10.49.14) combine the second of (10.51.3), with and , and the first of (10.49.3). Similarly for the other results.
Keywords:: spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.49.iii

10.49.14

$\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)=z^{n}\left(-\frac{1}{z}% \frac{d}{dz}\right)^{n}\frac{\mathop{\sin\/}\nolimits z}{z},$

$\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)=-z^{n}\left(-\frac{1}{z}% \frac{d}{dz}\right)^{n}\frac{\mathop{\cos\/}\nolimits z}{z}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, : integer and : complex variable
A&S Ref:: 10.1.25, 10.1.26
Permalink:: http://dlmf.nist.gov/10.49.E14
Encodings:: TeX, TeX, pMML, pMML, png, png

10.49.15

$\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=z^{n}\left(% \frac{1}{z}\frac{d}{dz}\right)^{n}\frac{\mathop{\sinh\/}\nolimits z}{z},$

$\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=z^{n}\left(% \frac{1}{z}\frac{d}{dz}\right)^{n}\frac{\mathop{\cosh\/}\nolimits z}{z}.$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer and : complex variable
A&S Ref:: 10.2.24, 10.2 25
Permalink:: http://dlmf.nist.gov/10.49.E15
Encodings:: TeX, TeX, pMML, pMML, png, png

10.49.16 $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)=(-1)^{n}\tfrac{1}{2}\pi z% ^{n}\left(\frac{1}{z}\frac{d}{dz}\right)^{n}\frac{e^{{-z}}}{z}.$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : base of exponential function, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.49.E16
Encodings:: TeX, pMML, png

§10.49(iv) Sums or Differences of Squares

Notes:: From (10.18.6), (10.47.3), and (10.47.4), ${\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{\mathsf{% y}_{{n}}\/}\nolimits^{{2}}}\!\left(z\right)=(\pi/(2z)){\mathop{M_{{n+\frac{1}{% 2}}}\/}\nolimits^{{2}}}\!\left(z\right)$ . Then apply (10.18.17). To derive (10.49.20) combine (10.47.12) and (10.49.18).
Keywords:: spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.49.iv

Denote

10.49.17 $s_{k}(n+\tfrac{1}{2})=\frac{(2k)!(n+k)!}{2^{{2k}}(k!)^{2}(n-k)!},$ $k=0,1,\ldots,n$ .

Symbols:: : : factorial, : integer, : nonnegative integer and $s_{k}(n)$
Permalink:: http://dlmf.nist.gov/10.49.E17
Encodings:: TeX, pMML, png

Then

10.49.18 ${\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{\mathsf{% y}_{{n}}\/}\nolimits^{{2}}}\!\left(z\right)=\sum_{{k=0}}^{n}\frac{s_{k}(n+% \frac{1}{2})}{z^{{2k+2}}}.$

Symbols:: $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, : integer, : nonnegative integer, : complex variable and $s_{k}(n)$
A&S Ref:: 10.1.27
Referenced by:: §10.49(iv)
Permalink:: http://dlmf.nist.gov/10.49.E18
Encodings:: TeX, pMML, png

10.49.19

${\mathop{\mathsf{j}_{{0}}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{\mathsf{% y}_{{0}}\/}\nolimits^{{2}}}\!\left(z\right)=z^{{-2}},$

${\mathop{\mathsf{j}_{{1}}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{\mathsf{% y}_{{1}}\/}\nolimits^{{2}}}\!\left(z\right)=z^{{-2}}+z^{{-4}},$

${\mathop{\mathsf{j}_{{2}}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{\mathsf{% y}_{{2}}\/}\nolimits^{{2}}}\!\left(z\right)=z^{{-2}}+3z^{{-4}}+9z^{{-6}}.$

Symbols:: $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind and : complex variable
A&S Ref:: 10.1.28--10.1.30
Permalink:: http://dlmf.nist.gov/10.49.E19
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

10.49.20 $\left(\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)\right)^{2% }-\left(\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)\right)^% {2}=(-1)^{{n+1}}\sum_{{k=0}}^{n}(-1)^{k}\frac{s_{k}(n+\frac{1}{2})}{z^{{2k+2}}}.$

Symbols:: $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer, : nonnegative integer, : complex variable and $s_{k}(n)$
A&S Ref:: 10.2.26
Referenced by:: §10.49(iv)
Permalink:: http://dlmf.nist.gov/10.49.E20
Encodings:: TeX, pMML, png

10.49.21

$\Big(\mathop{{\mathsf{i}^{{(1)}}_{{0}}}\/}\nolimits\!\left(z\right)\Big)^{2}-% \Big(\mathop{{\mathsf{i}^{{(2)}}_{{0}}}\/}\nolimits\!\left(z\right)\Big)^{2}=-% z^{{-2}},$

$\Big(\mathop{{\mathsf{i}^{{(1)}}_{{1}}}\/}\nolimits\!\left(z\right)\Big)^{2}-% \Big(\mathop{{\mathsf{i}^{{(2)}}_{{1}}}\/}\nolimits\!\left(z\right)\Big)^{2}=z% ^{{-2}}-z^{{-4}},$

$\Big(\mathop{{\mathsf{i}^{{(1)}}_{{2}}}\/}\nolimits\!\left(z\right)\Big)^{2}-% \Big(\mathop{{\mathsf{i}^{{(2)}}_{{2}}}\/}\nolimits\!\left(z\right)\Big)^{2}=-% z^{{-2}}+3z^{{-4}}-9z^{{-6}}.$

Symbols:: $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function and : complex variable
A&S Ref:: 10.2.27--10.2.29
Permalink:: http://dlmf.nist.gov/10.49.E21
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.48 Graphs 10.50 Wronskians and Cross-Products

§10.49 Explicit Formulas

Contents

§10.49(i) Unmodified Functions

§10.49(ii) Modified Functions

§10.49(iii) Rayleigh’s Formulas

§10.49(iv) Sums or Differences of Squares